GATE EC 2014 Set 4 — Question 53
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Signals & Systems → LTI Systems → Pole-Zero Stability & Causality
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Question
A stable linear time invariant (LTI) system has a transfer function . To make this system causal it needs to be cascaded with another LTI system having a transfer function . A correct choice for among the following options is
Correct answer
(B) s - 2
Solution
The transfer function is . The poles are at and .
For a stable system, the Region of Convergence (ROC) must include the axis. The possible ROCs are:
To make the system causal, the ROC must be to the right of the rightmost pole. This requires removing the pole at .
By cascading with , the pole at is cancelled:
.
The resulting system has a single pole at . For it to be both stable and causal, the ROC is , which includes the axis. Thus, is a correct choice.
For a stable system, the Region of Convergence (ROC) must include the axis. The possible ROCs are:
1. (unstable, anti-causal)
2. (stable, non-causal)
3. (unstable, causal)
Since the system is given as stable, its ROC is . To make the system causal, the ROC must be to the right of the rightmost pole. This requires removing the pole at .
By cascading with , the pole at is cancelled:
.
The resulting system has a single pole at . For it to be both stable and causal, the ROC is , which includes the axis. Thus, is a correct choice.
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